Weyl and Clifford Algebras

WEYL AND CLIFFORD ALGEBRAS.

VERA SERGANOVA

0.1. Weyl algebra. Let k be a ﬁeld and V be a ﬁnite-dimensional vector space over k. Let ω be a non-degenerate skew-symmetric bilinear form on V , i.e. ω(x, x) = 0 for any x ∈ V and for x 6= 0 there exists y such that ω(x, y) 6= 0.

Lemma 0.1. The dimension of V is even and there exists a basis x1, . . . , xn, y1, . . . , yn in V such that ω(xi, xj) = ω(yi, yj) = 0 and ω(yi, xj) = δij.

Proof. By induction in the dimension of V . Pickup a non-zero y1 ∈ V . Since ω is non-degenerate, there exists x1 ∈ V such that ω(y1, x1) = 1. Now let W be the ⊥ subspace generated by x1, y1. For any S ⊂ V let S = {x ∈ V | ω(x, s) = 0∀s ∈ S}. ⊥ ⊥ ⊥ ⊥ ⊥ Then dim x1 = dim y1 = dim V − 1, and dim W = dim(x1 ∩ y1 ) = dim V − 2. Now we have V = W ⊕ W ⊥, the restriction of ω on W ⊥ is non-degenerate and by ⊥ induction assumption we can choose a basis x2, . . . , xn, y2, . . . , yn in W satisfying the conditions of Lemma. Hence the statement.

A Weyl algebra Dω is by deﬁnition the quotient of the tensor algebra T (V ) by the ideal generated by x ⊗ y − y ⊗ x − ω(x, y) for all x, y ∈ V . It follows from Lemma 0.1 that Dω actually depends only on dim V = 2n and can be deﬁned as an associative k-algebra with generators x1, . . . , xn, y1, . . . , yn and relations

(0.1) [xi, xj] = [yi, yj] = 0, [yi, xj] = δij. Lemma 0.2. If V and W be ﬁnite-dimensional vector spaces with non-degenerate 0 skew symmetric forms ω and ω respectively. Then Dω⊕ω0 ' Dω ⊗ Dω0 .

Proof. The map T (V ⊕ W ) → Dω ⊗ Dω0 which maps (v, w) to v ⊗ 1 + 1 ⊗ w induces a homomorphism of associative algebras. Write Dω⊕ω0 = T (V ⊕ W )/I. By a straightforward check I lies in the kernel of this homomorphism, hence we have a homomorphism Dω⊕ω0 → Dω ⊗ Dω0 . To construct the inverse map note that the embeddings V,W ⊂ V ⊕ W induce the homomorphisms T (V ) → Dω⊕ω0 and T (W ) → Dω⊕ω0 , that can be pushed down to Dω → Dω⊕ω0 and Dω0 → Dω⊕ω0 . Since the images of Dω and Dω0 commute we obtain a homomorphism Dω ⊗ Dω0 → Dω⊕ω0

Lemma 0.3. Let x1, . . . , xn, y1, . . . , yn be a basis in V satisfying the conditions of a1 an b1 bn Lemma 0.1. Then the set of all monomials of the form x1 ··· xn y1 ··· xn for all non-negative integers a1, . . . , an, b1, . . . , bn form a basis of Dω. Date: April 11, 2012. 1 2 VERA SERGANOVA

Proof. By Lemma 0.2 and Lemma 0.1 it is suﬃcient to prove the statement in the case dim V = 2. This can be done by the diamond lemma using the lexicographical a b c d order on monomials x y x y ... .

Let R = k[X1,...,Xn]. Let xi denote the k-linear operator on R given by multipli- ∂ cation on Xi and ∂i = . There exists a unique homomorphism T (V ) → Endk(R) ∂Xi that maps xi to xi and yi to ∂i. This homomorphism respects the relations (0.1) and therefore induces a homomorphsim ρ : Dω → Endk(R). Thus, R is a Dω-module. p Exercise 1. Prove that if characteristic of k is p > 0, then ∂i = 0 and hence ρ is not injective. If characteristic of k is 0, then ρ is injective and M is a simple faithful Dω-module. 0.2. Diﬀerential operators. Let R be an arbitrary commutative k-algebra. Set D0(R) = EndR(R) = R and deﬁne Di(R) for i > 0 inductively by

Di(R) = {d ∈ Endk(R) | [d, D0(R)] ⊂ Di−1(R).

The identity [f, d1d2] = [f, d1]d2 +d1[f, d2] implies Di(R)Dj(R) ⊂ Di+j(R). Therefore [ D(R) = Di(R) i≥0 is a subalgebra in Endk(R). It is called the algebra of differential operators in R and its elements are called differential operators. The order of a differential operator d is the minimal i such that d ∈ Di(R). L Lemma 0.4. [Di(R),Dj(R)] ⊂ Di+j−1 and therefore Gr D(R) = Di(R)/Di−1(R) is commutative.

Proof. Proceed by induction in i and j. Let d1 ∈ Di(R), d2 ∈ Dj(R). For any f ∈ D0 we have [f, [d1, d2]] = [[f, d1], d2] + [d1, [f, d2]].

Since [f, d1] ∈ Di−1(R) and [f, d2] ∈ Dj−1(R), we obtain [f, [d1, d2]] ∈ Di+j−2(R) by induction assumption. Therefore [d1, d2] ∈ Di+j−1(R). Theorem 0.5. Let ω be a non-degenerate skew-symmetric form on 2n-dimensional vector space V . If char k = 0, then Dω ' D(k[X1,...,Xn]).

Proof. We have deﬁned already a Dω-module structure on k[X1,...,Xn]. We set

p m1 mn Dω = span{k[x1, . . . , xn]∂1 ··· ∂n | m1 + ··· + mn = p}. 0 p p−1 0 Then it is easy to check that [Dω,Dω] ⊂ Dω . Since Dω = k[x1, . . . , xn], we have p Dω ⊂ Dp(k[X1,...,Xn]). To prove equality proceed by induction. Assume d ∈ Dp(k[X1,...,Xn]). Then di = [d, xi] ∈ Dp−1(k[X1,...,Xn]) for all i ≤ n. Note p−1 that by induction assumption di ∈ Dω . Moreover, [[d, xi], xj] = [[d, xj], xi] hence [di, xj] = [dj, xi]. I leave it to you as an exercise (see home work 11.2) to check that 0 p 0 0 there exists d ∈ Dω such that di = [d , xi]. Then [d − d , xi] = 0 for all i ≤ n, WEYL AND CLIFFORD ALGEBRAS. 3

0 0 hence [d − d , f] = 0 for all f ∈ k[X1,...,Xn]. Therefore d − d ∈ k[x1, . . . , xn] and p d ∈ Dω. 0.3. The Clifford algebra. Now let char k 6= 2, V be a finite-dimensional vector space over k and g be a symmetric bilinear form on V .A Clifford algebra Cg is the quotient of the tensor algebra T (V ) by the ideal generated by x ⊗ y + y ⊗ x − g(x, y) for all x, y ∈ V . If x1, . . . , xn is an orthogonal basis in V with respect to the form g, then Cg is generated by x1, . . . , xn subject to relations 2 xixj = −xjxi, if i 6= j; 2xi = g(xi, xi).

Examples. If g = 0, then Cg = Λ(V ). If dim V = 1 and g 6= 0, then Cg is either k ⊕ k or some quadratic extension of the field k. In particular, if k = R, then Cg is either R ⊕ R or C. If k = R and g is a negative definite form on a 2-dimensional space, then Cg is isomorphic to the ring H of quaternions. Indeed, for a suitable basis {i, j} in V we 2 2 have i = j = −1, ij = −ji. Then 1, i, j, k = ij form a basis of Cg over R and satisfy the quaternion’s relations. It is easy to see that Cg is a superalgebra, i.e. Cg has a Z2-grading that is induced by the natural Z-grading on T (V ). In this grading all x ∈ V are odd. Recall the definition of super tensor product (Lang XVI.6). If A and B are two superalgebras, then A ⊗su B coincides with A ⊗ B as a vector space, and multiplication is defined by the formula (a ⊗ b)(a0 ⊗ b0) = (−1)deg(b)deg(a0)aa0 ⊗ bb0. Lemma 0.6. Let V and W be vector spaces equipped with symmetric forms g and 0 g respectively. Then Cg⊕g0 = Cg ⊗su Cg0 The proof is similar to the proof of Lemma 0.2 and is left to the reader. Since g has an orthonormal basis the above lemma implies n Corollary 0.7. If dim V = n, then dim Cg = 2 . If x1, . . . , xn is an orthogonal basis in V , then the set {xi1 ··· xis | i1 < ··· < is} is a basis of Cg. Lemma 0.8. Let k be algebraically closed, W ⊂ V , dim V = dim W + 1 and dim W is even. Let g be a non-degenerate symmetric form on V such that the restriction g0 of g on W is also non-degenerate. Then Cg ' Cg0 ⊕ Cg0 .

Proof. One can choose an orthogonal basis x1, . . . , x2m+1 in V such that x1, . . . , x2m is a basis of W . Then θ = x1 ··· x2n+1 is a central element in Cg since xiθ = θxi for 2 all i ≤ 2m + 1. After a suitable normalization of xi, we can obtain θ = 1. Then ± 1±θ + − e = 2 are two central idempotents, e +e = 1, and it follows by a straightforward ± check that e Cg ' Cg0 .

Assume now that dim V = 2m and there exists a basis x1, . . . , xm, y1, . . . , ym in V such that g(xi, xj) = g(yi, yj) = 0, g(xi, yj) = δij. Let W be the subspace spanned 0 by x1, . . . , xm and W be the subspace spanned by y1, . . . , ym. As in the case of the 4 VERA SERGANOVA

Weyl algebra we can define a structure of a Cg-module on the exterior algebra Λ(W ). For any x ∈ W we define σ(x) ∈ Endk(Λ(W )) as the wedge multiplication by x on the left. If y ∈ W 0 we define σ(y)(x) = g(y, x) for any x ∈ W and extend it to σ(y) ∈ Endk(Λ(W )) by the following rule y(α ∧ β) = y(α) ∧ β + (−1)deg(α)α ∧ y(β).

Exercise 2. Prove that the above map defines an isomorphism σ : Cg → Endk(Λ(W )). Remark. For an associative superalgebra A denote the commutator by the formula [a, b] = ab − (−1)deg(a)deg(b)ba. In particular, a superalgebra is commutative if [a, b] = 0 for all a, b ∈ A. As in the usual case we can define the superalgebra of differential operators D(A) for any commutative superalgebra A. Note that Endk(A) has a natural Z2 grading with even part Endk(A0¯) ⊕ Endk(A1¯) and odd part Homk(A0¯,A1¯) ⊕ Homk(A1¯,A0¯). The definition of the superalgebra of differential operators is the same as the defintion of the algebra of differential operators with understanding that [·, ·] defines the supercommutator. Then under above assumptions we have Cg ' D(Λ(W )). Corollary 0.9. Let k be algebraically closed and g be a non-degenerate symmetric bilinear form on V . If dim V = 2m, then Cg ' Mat2m (k) and if dim V = 2m + 1, then Cg ' Mat2m (k) ⊕ Mat2m (k).